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Greedy approximation in certain subsystems of the Schauder system. (English) Zbl 1177.42025

A Schauder basis for a Banach space \(X\) is a countable set \(\Psi:=\{\psi_n\mid n\in\mathbb{N}\}\subset X\) with respect to which each \(f\) in \(X\) can be represented by a unique series \(\sum_n \mathcal{C}_n(f)\psi_n\) that converges to \(f\) in the norm of \(X\).
Let \(\sigma:\mathbb{N}\rightarrow\mathbb{N}\) be a bijection for which \(|\mathcal{C}_{\sigma(n)}(f)|\geq|\mathcal{C}_{\sigma(n+1)}(f)|\), then \[ G_m(f)=\sum_{n=1}^m\mathcal{C}_{\sigma(n)}\psi_{\sigma(n)} \] is the \(m\)th greedy approximant of \(f\) with respect to the basis \(\Psi\) and the permutation \(\sigma\).
It is known, that there are functions in \(L^p[0,1]\) (\(1\leq p<2\)), for which a sequence of corresponding greedy approximants diverges in measure.
Although the greedy aproximants of \(f\) may diverge, the author proves that there always will be a continuous function \(g\), arbitrarily close to \(f\) in measure, such that the sequence of greedy approximants of \(g\) converges uniformly to \(g\).

MSC:

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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